How does The Fools Guild make its money?

We have to assume that fools, jesters and clowns are paid for their services in the same way that musicians are paid. In Soul Music, we find that performing arts on the streets of Ankh-Morpork for money is a thing, which requires a license and guild membership.

He'd found a place in one of the main squares, tuned up and played. No-one had taken any notice, except sometimes to push him out of the way as they hurried past and, apparently, to nick his bowl. Eventually, just when he was beginning to doubt that he'd made the right decision in coming here at all, a couple of watchmen had wandered up. (...)

'Sounds like garglin' with gravel to me,' said the one identified as Nobby. 'You got a licence, mate?'

'Licence?' said Imp. 'Very hot on licences, the Guild of Musicians,' said Nobby. 'They catch you playing music without a licence, they take your instrument and they shove-'

'Now, now,' said the other watchman. 'Don't go scaring the boy.'

In Making Money, we see that the Fool's Guild also trains and educates otherwise rational people to get into the silly costumes provided and act as hosts and guides at theme parks. Supposedly the theme parks pay those people.

The same wiki I linked above also states that guild members learn about a dark art called Sloshi. It is a martial art described as "clowning without the moderation". There are very violent records of its usage, with clowns being able to disperse death and grave injury. In the past this art has been used in the defense of Ankh-Morpork against barbarians. The wiki on the guild goes on to say that:

This may shed further light on why the Fools' Guild shares a party wall with the Assassins' Guild, and why this has been used for clandestine contact between the two Guilds.

So supposedly you could hire mercenary clowns as well.

I think you're underestimating the most obvious source of income for the Guild, providing (for a fee) trained fools for kings and queens. In Wyrd Sisters we encounter a fool who's working in a kingdom with a population of a few thousand and there are, we learn, literally hundreds of these small kingdoms dotted around the Ramtops as well as dozens more on the Sto Plains.

Any self-respecting king or queen needs a fool, and where better to get one than by requesting one from the world-renowned Fool's Guild of Ankh-Morpork?

There were little kingdoms all over the hubward slopes of the Ramtops. Every narrow valley, every ledge that something other than a goat could stand on, was a kingdom. There were kingdoms in the Ramtops so small that, if they were ravaged by a dragon, and that dragon had been killed by a young hero, and the king had given him half his kingdom as per Section Three of the Heroic Code, then there wouldn’t have been any kingdom left. There were wars of annexation that went on for years just because someone wanted a place to keep the coal.

_{^{Lord and Ladies}}

and

There’d been a lot of that sort of thing, Vimes had been surprised to learn. The little kingdoms of the Sto Plains were forever scrounging spare royalty off one another.

_^Jingo

The Discworld Companion (Turtle Recall edition) confirms this and offers a considerable amount of detail about the Guild's background and financial wellbeing, albeit much of this is to be considered hearsay

One of the more recent Ankh-Morpork Guilds, although like the ASSASSINS’ GUILD and Unseen University it is a major exporter of its graduates and has ancient origins among the circus fraternity.

and

Over the years the Guild’s influence of all sorts of foolery is such that there soon were very few fools in civilised countries who were not graduates. It has been suggested that the new Guild became nothing more than a vast spy network, sending back to the House of Mirth snippets of political information which were used by the guild council to become enormously rich.

We must make it CLEAR that the Guild’s vast wealth accrues from PROWESS WITH THE CUSTARD PIE, CAREFUL CONTROL OVER EXPENDITURE and other Foolish activities. Dr Whiteface is no more than a hard-working administrator, and certainly NOT the cruel and devious international manipulator that UNSUBSTANTIATED RUMOUR makes him out to be. Anyone suggesting otherwise can expect a visit from the Jolly Good Pals in VERY SHORT ORDER.

The Guild also supplies other workers in the entertainment industry.

Nevertheless, most of the Disc’s fools, jesters, minstrels, idiots and mimes herald from Ankh-Morpork’s Guild. Indeed, in 1788, it bought out La Sorbumme and now runs that as a summer school.

Far down the canon list but the Discworld: Ankh Morpork board game hints at a sinister extortion element to the guild.

Both the 'Dr Whiteface' and 'Fool's Guild' cards require an opposing player to give up 5 dollars or accept a permanent reduction of their hand size to 4 (rather than the default 5). In the early game this can be crippling and is one of the most aggressive actions you can take against a player who has recklessly or unluckily ended up broke.

Thematically this suggests that the Guild clowns make money in Ankh Morpork (like the Canting Crew) by forcing people to pay them or, if they refuse, by sticking around indefinitely.

This fits entirely with Pratchett's negative view of clowns. To take a quote from Men at Arms:

No clowns were funny. That was the whole purpose of a clown. People laughed at clowns, but only out of nervousness. The point of clowns was that, after watching them, anything else that happened seemed enjoyable.

@Valorum's answer touched on this, but didn't really state it outright (probably because there's no real solid source besides the Discworld Companion).

Politics and spies, oh my!

All quotes come from The Discworld Companion.

First of all, they started out already having money.

It was in fact Monsieur Jean-Paul Pune who founded the Guild ... M. Pune was a wealthy man.

And then...

It has been suggested that the new Guild became nothing more than a vast spy network, sending back to the House of Mirth snippets of political information which were used by the guild council to become enormously rich.

In addition, of course:

There was already an ancient tradition on Foolery ... confidants and advisers, whose role was tell the king things that he really needed to know

Note, however, that most of this info is presented in the Companion in a rather tongue-in-cheek manner, likely because the author's guestroom still needs a little more tidying up.

We must make it CLEAR that the Guild’s vast wealth accrues from PROWESS WITH THE CUSTARD PIE, CAREFUL CONTROLOVER EXPENDITURE and other Foolish activities. Dr Whiteface is no more than a hard-working administrator, and certainly NOT the cruel and devious international manipulator that UNSUBSTANTIATED RUMOUR makes him out to be. Anyone suggesting otherwise can expect a visit from the Jolly Good Pals in VERY SHORT ORDER.